reserve r,r1,g for Real,
  n,m,k for Nat,
  seq,seq1, seq2 for Real_Sequence,
  f,f1,f2 for PartFunc of REAL,REAL,
  x for set;
reserve r,r1,r2,g,g1,g2 for Real;

theorem
  f1 is divergent_in-infty_to-infty & (ex r st left_open_halfline(r) c=
dom f /\ dom f1 & for g st g in left_open_halfline(r) holds f.g<=f1.g) implies
  f is divergent_in-infty_to-infty
proof
  assume
A1: f1 is divergent_in-infty_to-infty;
  given r1 such that
A2: left_open_halfline(r1)c=dom f/\dom f1 and
A3: for g st g in left_open_halfline(r1) holds f.g<=f1.g;
A4: now
    let r;
    consider g being Real such that
A5: g<-|.r.|-|.r1.| by XREAL_1:2;
    take g;
    0<=|.r1.| & -|.r.|<=r by ABSVALUE:4,COMPLEX1:46;
    then -|.r.|-|.r1.|<=r-0 by XREAL_1:13;
    hence g<r by A5,XXREAL_0:2;
    0<=|.r.| & -|.r1.|<=r1 by ABSVALUE:4,COMPLEX1:46;
    then -|.r1.|-|.r.|<=r1-0 by XREAL_1:13;
    then g<r1 by A5,XXREAL_0:2;
    then g in {g2: g2<r1};
    then g in left_open_halfline(r1) by XXREAL_1:229;
    hence g in dom f by A2,XBOOLE_0:def 4;
  end;
  now
    dom f/\dom f1 c=dom f by XBOOLE_1:17;
    then
A6: dom f/\left_open_halfline(r1)=left_open_halfline(r1) by A2,XBOOLE_1:1,28;
    dom f/\dom f1 c=dom f1 by XBOOLE_1:17;
    hence
    dom f/\left_open_halfline(r1) c=dom f1/\left_open_halfline(r1) by A2,A6,
XBOOLE_1:1,28;
    let g;
    assume g in dom f/\left_open_halfline(r1);
    hence f.g<=f1.g by A3,A6;
  end;
  hence thesis by A1,A4,Th73;
end;
